I recently made a math demo for Kneser
Graphs. These are basically pretty graphs based on
dominoes, but I was long stymied on how to draw these nicely until Doug
West gave me the hint "permutations". One permutation
I found was 13527486, so I looked that up, and found that this permutation
is heavily used in church bell ringing. For example, Matthew Sorrell's "Mathematics
of Church Bell Ringing" (a fascinating paper) lists this permutation
as Plain Bob. In bell ringing, eight bells are rung in pairs, with every
pair represented. Same as double-7 dominoes without the doubles.

Bob Abbott writes: I just added something to my site that I think is the best
writing I've done about cornfield and other large mazes. It is Mazes We Visited —
Summer, 2007. It starts out like a journal, but then I get into
reviews (good and bad, but mostly bad) of various mazes. I explain what
they do right and what they do wrong, and I give my own opinion of what
a maze should be. I have a brief history of the rise and fall of wooden
fence mazes, and I think I have a good explanation of why they
disappeared. I also warn that cornfield mazes could suffer the same
fate. Also, there is a description of an implementation of Ed Pegg's
UP-DOWN multi-state maze.

Yang Chao: I recently wrote a little program "Toads
and Frogs Puzzle",
which is a generalization of Sam Loyd's classic puzzle "fore and aft".
With this little tool, it is convenient to explore puzzles of this type. The "fore
and aft" type puzzle resembles the sliding blocks puzzle with extra permission
that a block can jump over another, so it is interesting to play with.

Nucleus

Bryce Herdt: After seeing the hexagonal Black Box on your site, I decided
to push my hexagonal Magnets-like
puzzle, called Nucleus. The groups of three
circles are either protons or neutrons, the black circle is a gluon, and the
uncolored circles are quarks. Quarks are either red, green, or blue, and though
in neutrons they cannot be distinguished, same-colored quarks never touch among
protons. Of course, each proton consists of one red, one green, and one blue
quark. Given the numbers of distinguishable quarks in some lines, determine
the placements of all the protons.

The new Times Square ball that
will be dropped for the new years has a very mathematical shape. The video showing
the half-ton ball in action is well worth a watch.

The Rolling Ball Web

A compendium of rolling ball sculptures and clocks is available at the Rolling
Ball Web. It seems to need an update.

Sums of Three Cubes

The list of unsolved values x under 500 (not of the form 9n±4) for
x=a3+b3+c3 is
down to 33 42 74 114 165 390. Results are at the bottom of Joerg
Jahnel's Research page. Some history is at Daniel
Bernstein's page, and
in MathWorld's
Cubic Number entry.
For example, for x=52, one answer for a,b,c is 60702901317, 23961292454, and
-61922712865.

Sums of 4 Quartics

More solutions to a4=b4+c4+d4 are
known. A full list is at the bottom of http://www.euler413.narod.ru/.
One example is 8738221214 = 7693212804 + 6067108714 +
5584244404.
Recent discoveries found with Robert
Gerbicz's
program.

Posted earlier: In Morpion, starting with a 36-point cross shape, you
get 1 point for each additional 5-in-a-row. A
score of 79 points in Morpion was recently acheived. UPDATE:
Much higher scores are listed at euler.free.fr.
UPDATE 2, from Michael Quist: You recently posted results
for Morpion Solitaire. Actually, the lower and higher records are for two
slightly different variants of Morpion Solitaire: in one variant ("disjoint"),
two lines in the same direction may not share an endpoint; in the other
("touching"), they may. (The lower score, of course, is for the
disjoint variant.) Demaine et al. wrote a Morpion
paper in which they define these and other variants and prove upper and lower
bounds for record scores.

Peter Grabarchuk: I was inspired by puzzle The
Four Queens by Martin Gardner
and
have made a related puzzle. Arrange 4 queens and 2 bishops on a 10x10 board
so that every square is either attacked or occupied. Also, I'm happy to
present you my personal website PeterPuzzle.com, launched past week.

Brian Trial: Enclosed is an order 60 SPSR 705x3525 with an integer aspect
ratio of 5. [Many similar results are at squaring.net.]
Picture.

Material added 3 November 2007

Hexagonal Black Box

Eric Solomon: Franjos released
BLACK BOX + at Essen. This contains the original game and the new hexagonal
version which is more demanding. The set also contains a book of puzzles with
solutions. The game is obtainable in the USA from www.funagain.com.
I attach one difficult puzzle.
The solution is, of course, unique.

In the June 1959 Scientific American, Martin
Gardner presented Eleusis, the card game that uses inductive reasoning. Later,
in October 1977, he presented an improved and expanded version of the game.
Now there is a third version—called Eleusis Express. It is smaller and
faster than the full version of Eleusis, and it has the possibility of being
a fun, social game. You can read about it at logicmazes.com.

17orBust: 33661•2^7031232+1
is prime! The double checking effort uncovered this new prime showing once
again why it is so important to conduct these re-testing efforts. Most re-tests
conclusively verify the original result but if the first test had even a
single memory error during the computation, there's a chance a prime could
have been missed. That's what likely happened here. The discoverer, Sturle
Sunde is a giant in the prime searching world. He is the #2 overall contributor
to SB and also the #4 all-time contributor to GIMPS! [Now the 10th largest
known prime.]

Toby Gottfried: I may not be the first to have had this idea. Rubik's
Grid. 5x5 grid of numbers
1-5 in 5 colors Object: make each line read 1 2 3 4 5 in same color (or
each column, or one of each color in each line, or ....) Available moves:
swapping two adjacent pairs of numbers.

Ancient grid pattern

Discovery
Channel: Deep in the heart of northern Syria, close to the banks of
the Euphrates River, archaeologists have uncovered a series of startling
11,000-year-old wall paintings and artifacts. [Square grid, with brown,
black, and white colored squares. Related puzzle. What word relating to
squares has the letters of SYRIA, in order?]

Material added 24 October 2007

Alex Smith Proves the 2,3 Turing Machine is Universal

As reported at New
Scientist, Nature,
and at WolframScience,
20-year-old student Alex Smith has proven that a particular 2,3 Turing machine
is a universal computer, as first conjectured by Stephen Wolfram. For his
impressive proof, Alex collects a $25,000 prize.

Material added 21 October 2007

Euphoric Pentahexes

Col. George Sicherman: A euphoric polyform is compatible with all polyforms
of some order in the same class. The Five
Euphoric Pentahexes are each compatible
with all 82 hexahexes. [Wow. Just about all the careful discoveries at Polyform
Curiousities are incredible.]

Peter Esser: With three colors you get 59 different cubes. Discarding the
three cubes with only one color gives 54 cubes allowing for a 3x3x6 box with
same colors at opposite faces. [Peter's
Polyform page is excellent.]

Order 3 Projective Plane

Bob Harris: Here's my attempt at drawing the projective plane. It doesn't
quite have the symmetry that your drawing does. Might look more appealing
if the central 3x3 square were rotated 22.5 deg ccw or 67.5 deg cw. And/or
the 3C64 line could be C643 to balance C17B (ditto for 0547 relative to 5618).
[Bob runs the excellent logic puzzle site bumblebeagle.org.]

Material added 22 September 2007

Numb3rs Site and Trip

I'll be at the Numb3rs
Season Premiere on Monday. The broadcast premiere in the US is Friday.
First time I've ever been to anything like this. I helped with the new numb3rs.wolfram.com site,
particularly the page of MathPlayer demonstrations. There's also a link at cbs.com.

Word Search

Place as many words as possible into a 15x15 grid for the Words
Search Al Zimmermann contest. A total of $400 in prizes.

77 Moves Enough for Order-4 Rubik's Cube

Jean-Charles Meyrignac: That's the new benchmark. Brusce Norskog has
the analysis.

Projective Plane

I recently made a cool picture of an order-3 projective plane. I'm still
trying to figure out the best way to draw it - the current usage of 6 is a
flaw.

George Miller wins prize for puzzle-making video

From dimensionprinting.com, George Miller won
an award for his video how "3D printing revolutionized
his puzzle business" (puzzlepalace.com).
George explains: "In one
occasion, my friend Oskar from Holland emailed me a design, which my Santa
Claus machine printed overnight, so I could ship it to his customer in New
York the next morning". George and Oskar have several
new designs to be
amazed by.

Material added 16 September 2007

Flatland: The Competing Versions

There are two versions of Edwin Abbott's Flatland:
The Book coming out soon. The first is a 95-minute version, Flatland:
The Film, put out by Ladd Ehlinger, Jr. I bought a copy ($22
at Amazon)
and watched it, and will give it 3 stars. Very faithful to the original book,
with excellent visuals and good math. The voice acting was excellent, even
though most of the actors are unknowns. Looking down on the 2-D creatures,
their heartbeat and actual thoughts can be seen. On the downside, at the
beginning, there is frequent "silent
film" style
title narration that tries to help explain things. Various reviews are at
imdb and amazon.

Then there is the 30-minute Flatland:
The Movie. It stars Martin Sheen and Kirsten Bell, and cannot yet be
purchased ($30). The whole
world seems heavily influenced by fractals. I haven't seen the film yet,
but I have watched the trailer. This production obviously had a bigger budget.
One review is at imdb.

A puzzle from 2002 was The Grand Vizier, by Pavel Curtis, based on Penrose
tiles. A very pretty puzzle. He has a stash
of copies left, if anyone would
like to buy one.

Largest Consecutive Factorizations

Can you completely factor 447295839 · 27061 − 3,
and the next four numbers? Donovan Johnson managed it, setting the record
for 5 completely factored consecutive numbers. Jens Kruse Andersen is maintaining
a Consecutive
Factorizations page, which has a lot of clever ideas. Virtually all of
the records are from the past 2 months.

Material added 19 August 2007

Disdyakis - A Mathematical Tungsten Carbide Ring

I've lately been fascinated with Tungsten
Carbide rings. They are as heavy as gold, almost indestructible, almost
unscratchable, and inexpensive. One of the best sources is Trueman
Jewelry.
I bought 30 different tungsten carbide rings from them for around $600. I
was very pleased with their quality. Now, I'm making a custom order, based
on the Disdyakis
Triacontahedron. I've recoved caps, so that each ring has 60 triangles.
If you'd like to buy one, I'll need your ring size, address, and $20 (cost
of shipping included. Add $5 for overseas shipping). You can pay me via
the Paypal link to the right. The money can wait, but I'll
need your ring size by September 5th. Any jeweler can help
you find your ring size. Sometime in November, I'll get the 80 or so Disdyakis
rings, and ship them all out. Interested? Send
me your ring size.

I've written a column about the Social
Golfer Problem. I even include a puzzle this time around. A less obvious
puzzle is the minimum number of clues to give a unique answer. Sixteen
social golfers (A to P) play in foursomes over five days. After the first
day, they pick out the tee-times on days where their schedules are tight.
Fill out the rest so the each golfer plays once each day, and plays exactly
once with every other golfer.
day1 day2 day3 day4 day5
9am ABCD GIP M IH G
1pm EFGH N FOD EJ JOB
3pm IJKL CEL PLH AK AM
5pm MNOP DK NK B CF

John Conway developed a beautiful sliding puzzle a few years back. Consider
13 lines going throught 13 points: {{0,1,3,9}, {1,2,4,10}, {2,3,5,11},
{3,4,6,12}, {4,5,7,0}, {5,6,8,1}, {6,7,9,2}, {7,8,10,3}, {8,9,11,4}, {9,10,12,5},
{10,11,0,6}, {11,12,1,7}, {12,0,2,8}}. Put numbered counters on 1-12. Now,
shift any digit to the blank spot, and swap the other two digits on that line.
This simple process underlies deep
group theory. An M13 applet is
available, for trying the puzzle. This is basically a puzzle on the ({0,1,3,9}+n
mod 13) switching system. The {1,2,5,15,17}-21 might make a similarly nice
puzzle, with 20 counters. Move one to the empty spot, then rotate the other
three.

Multiples of single digits

The number 24 is the product of single digits, in two different ways: 4×6
or 3×8. 3256 has consecutive pairs 12, 24, 45, and each is the product
of single digits (4×8, 5×5, and 7×8).
Arrange the digits of 1-9 so that every consecutive pair is the product
of single digits.

A classic problem asks for the maximum overhang of dominoes over the edge
of a table. The paper Maximum
Overhang gives a number of new solutions.

Self-Referential Phone Numbers

376-9767 is an interesting phone number. If you add pairs in sequence, 376,
9767, 10143, 19910, 30053, 49963, 80016, 129979, 209995, 339974, 549969, 889943,
1439912, 2329855, 3769767 -- you eventually get the phone number back. There
are 3 other 7-digit phone numbers with this property.

Self-Referential Sentences

A lot of self-referential sentences are at http://home.att.net/~rhhardin/selfref.html.
This sentence contains three a's, three c's, two d's, twenty seven e's, four
f's, two g's, ten h's, eight i's, thirteen n's, six o's, ten r's, twenty five
s's, twenty three t's, three u's, three v's, six w's, three x's, and four y's.
Many others are listed.

Dick Saunders Jr.: What date will you turn over 3 Million Visitors? I will
guess July 4, 2007. [Ed - Hmmm... about 2993900 right now. And I'm back from
a week long vacation. Jean-Charles Meyrignac: As of 8 August,
you had 3,004,441 visitors.]

Material added 29 July 2007

New Rolling Cube Puzzle

Robert Abbott: There have been some
new developments in Rolling Cube mazes and I thought you'd want to write about
them in mathpuzzle.com. I added a description of these developments on my rolling
cube page. At an up-coming conference, there will be a paper (which
can be downloaded from my site) on the computational complexity of rolling
cube puzzles. This paper also includes a revolutionary new rolling cube maze,
and this maze is also on my site. [Bob's Sliding
Door Maze has also been reposted.]

Eternity II Puzzle

The 2 million dollar Eternity
II puzzle has been released. The Yahoo
Eternity II Group has lots of
mathematical discussion about it. It is essentially an edge-matching
puzzle, long championed by Kadon
Enterprises. Unlike a Kadon creation, though, the Eternity II puzzle
was designed to be fantastically difficult to solve. The previous Eternity
puzzle was solved by readers of this website -- but they had to create
new solving methods to conquer it. At Eternity2.net,
a distributed solving program by Dave Clark is available.

The results of the Man-Machine Poker Championship are in. In the story write-up
(NYT, Herald
Tribune), humans win. In related news, the Chinook system, under lead
programmer Jonathan
Schaeffer, has finished a complete analysis of Checkers. An earlier version
of Chinook was beaten by Dr
Marion Tinsley, considered the world's greatest
checkers player.

Tony Fatseas has put together www.samuelloyd.com a
beautiful site devoted to Sam Loyd. The Encyclopedia of Puzzles is there, and
high-quality reprints of some of Sam's best-known puzzles are there. Elsewhere,
you can also now buy a reprint of Loyd's
Encyclopedia, and Jerry Slocum's analysis of the true history of both the
15
Puzzle and Tangrams.

Puzzleup 2007

Emrehan Halici's puzzle competition at puzzleup.com will begin on August
1.

Dave Dyer: I think you might like Cameron Browne's new game, Truchet. I'm
in the process of adding it to Boardspace.net, so it will be playable online
sometime soon.

3 Million Visitors

Dick Saunders Jr.: What date will you turn over 3 Million Visitors? I will
guess July 4, 2007. [Ed - Hmmm... about 2993900 right now. And I'm back from
a week long vacation.]

Material added 07 07 07

Cloverfield, JJ Abrams, and Ethan Haas

The Untitled JJ Abrams project,
ultimately a SF movie now codenamed Cloverfield (debuts Jan 8, 2008), has a
puzzle site of sorts, ethanhaaswasright.com.
So far, the puzzles are attractive but tedious. A trailer is in front of the
Transformers movie, I've heard.

Would you like to make a challenging maze with many vortices? The paper by
Jie Xu and Craig S. Kaplan, Vortex
Maze Construction, gives an excellent overview of the problem, lots of
insight, and lots of mazes.

The Angel Wins

On an infinite 2D board, a chess piece called the Angel can move up to n spaces
each term. A piece called the Devil destroys one square each turn. If the Devil
can surround the Angel with destroyed squares, the Devil wins. Who wins? in
a recent paper, Peter Gacs argues that The
Angel Wins.

Alice is Lost

Eric Harshbarger has made an online puzzle series called Alice
is Lost, which
has an Alice in Wonderland motif.

Bloxorz is a fairly nice flash game with rolling block mazes, as championed
by Robert Abbott of logicmazes.com.

Walking and Hopping Toys

Can an object walk down a mild ramp with no gears or batteries? peter Steinkamp
shows many varieties of what he calls Unpowered
Walkers and Hoppers.

3 Million Visitors

Dick Saunders Jr.: What date will you turn over 3 Million Visitors? I will
guess July 4, 2007. [Ed - Hmmm... about 2969000 right now. Apologies for the
length of the break. In 1991 I
bought a house in Colorado for $17,777.77. Now on 7/7/7, I've closed on it
(for more), and moved most of the things in it to my home here in Champaign
IL. So I've done lots of packing, unpacking, cleaning, and truck driving. My
mom was living in the Colorado house, and now doing comfortably in an assisted
living facility.]

Site Goals

Martin Gardner celebrates math
puzzles and
Mathematical Recreations. This site aims to do the same. If you've made
a good, new math puzzle, send it to ed@mathpuzzle.com.
My mail address is Ed Pegg Jr, 1607 Park Haven, Champaign, IL 61820.
You can join my moderated recreational mathematics email list at http://groups.yahoo.com/group/mathpuzzle/.

All material on this site is copyright
1998-2007 by Ed Pegg Jr. Copyrights
of submitted materials stays with the contributor and is used with
permission.